# [time 659] Cylindrical But Locally Lorentzian Universes

Stephen P. King (stephenk1@home.com)
Thu, 02 Sep 1999 22:56:09 -0400

Hi All,

My friend Paul Hanna pointed me to this page. It is very similar to
what he is working on!

Onward,

Stephen
Cylindrical But Locally Lorentzian Universes

## Cylindrical But Locally Lorentzian Universes

```A three-dimensional space that is everywhere locally Eulcidean and
yet cylindrical in all directions can be constructed by embedding the
three spatial dimensions in a six-dimensional space according to the
parameterization

x1 = R1 cos x/R1       x2 = R1 sin x/R1
x3 = R2 cos y/R2       x4 = R2 sin y/R2
x5 = R3 cos z/R3       x6 = R3 sin z/R3

so the spatial Euclidean line element is

dx1^2 + dx2^2 + dx3^2 + dx4^2 + dx5^2 + dx6^2 = dx^2 + dy^2 + dz^2

giving a Euclidean spatial metric in a closed three-space with total
volume (2*pi)^3*R1*R2*R3.  It's been suggested (by Klaus Kassner,
among others) that we can subtract this from an ordinary temporal
component to give an everywhere-locally-Lorentzian spacetime that
is cylindrical in the three spatial directions, i.e.,

ds^2 = c^2 dt^2 - (dx^2 + dy^2 + dz^2)

However, this "subtraction" doesn't seem well motivated.  One way
of providing a motivation, and in the process making the universe
cylindrical in ALL directions, temporal as well as spatial, would
be to embed the entire 4D spacetime into a space of 8 dimensions,
2 of which are purely imaginary, like this

x1 =   R1 cos x/R1       x2 =   R1 sin x/R1
x3 =   R2 cos y/R2       x4 =   R2 sin y/R2
x5 =   R3 cos z/R3       x6 =   R3 sin z/R3
x7 = i R4 cos t/R4       x8 = i R4 sin t/R4

leading (again) to a locally Lorentzian 4D metric

(ds)^2  =  (dx)^2 + (dy)^2 + (dz)^2 - (dt)^2

but now *all four* of the dimensions x,y,z,t are periodic.  So
here we have an everywhere-locally-Lorentzian manifold that is
closed and unbounded in every spatial and temporal direction.
(Obviously this manifold contains closed time-like worldlines.)

This reminds me a bit of Stephen Hawking's recent attempts to
describe a universe that is finite but unbounded in time as well
as space.  He too invokes an imaginary time dimension, but aside
from that I don't think his model is related to the locally-flat
cosmology described here.  Anyway, the causal structure of such
a universe is interesting.

We might imagine that a flat, closed, unbounded universe of this
type would tend to collapse if it contained any matter, unless a
non-zero cosmological constant is assumed.  On the other hand, I'm
not sure what "collapse" would mean in this context.  It might
mean that the R parameters would shrink, but R is not a dynamical
parameter of the model.  The 4D field equations operate only on
x,y,z,t.  Also, any "change" in R would imply some meta-time
parameter T, so that all the R coefficients in the embedding
formulas would actually be functions R(T).

It seems that the flatness of the 4-space is independent of the
value of R(T), and if the field equations are satisfied for one
value of R they would be satisfied for any value of R.  But I'm
not sure how the meta-time T would relate to the internal time
t for a given observer.  It might require some "meta field
equations" to relate T to the internal parameters x,y,z,t.
Possibly these meta-equations would allow (require?) the value
of R to be "increasing" versus T, and therefore indirectly
versus our internal time t = f(T), in order to achieve stability.

On the general method of embedding a locally flat n-dimensional
space in a flat 2n-dimensional space, I wonder if every orthogonal
basis in the n-space maps to an orthogonal basis in the 2n-space
according to a set of formulas formally the same as those shown
above.  If not, is there a more general mapping that applies to
all bases?

By the way, the above totally-cylindrical spacetime has a natural
expression in terms of "octonion space", i.e., the Cayley algebra
whose elements are two ordered quaterions

x1 = i R1 cos x/R1       x2 = i R1 sin x/R1
x3 = j R2 cos y/R2       x4 = j R2 sin y/R2
x5 = k R3 cos z/R3       x6 = k R3 sin z/R3
x7 =   R4 cos t/R4       x8 =   R4 sin t/R4

Thus each point (x,y,z,t) in 4D spacetime represents two quaterions

q1 = x1 + x3 + x5 + x7

q2 = x2 + x4 + x6 + x8

To determine the absolute distances in this 8D space we again consider
the eight coordinate differentials, exemplified by

d x1  =  i R1 (-sin(x/R1)) (1/R1) (dx)

(using the rule for total differentials) so the squared differentials
are exemplified by

(d x1)^2  =  - sin^2(x/R1) (dx)^2

Adding up the eight squared differentials to give the square of the
absolute differential interval leads again to the locally Lorentzian
4D metric

(ds)^2  =  (dt)^2 - (dx)^2 - (dy)^2 - (dz)^2

```